The more photosites, the higher the resolution. The bigger the photosites, the more sensitive they are and therefore the less noisy they are. It's up to the designers to balance resolution and sensitivity for their designs.

It's possible to take a high resolution image and reduce the noise by averaging neighbouring pixels which also reduces resolution. So, to some extent, you can do the trade off in software. You still lose out with very small photosites because the borders between sites are a fixed width, meaning a bigger percentage of the sensor isn't detecting light. Also, reading out more pixels takes longer which can limit camera frame rates and use more power, causing issues with overheating for video applications.

Bigger sensor means more light capture. No increase in megapixel count would result in lower noise. Increase in megapixel count results in similar noise (smaller though) but higher detail.

Larger sensors also give more depth-of-field (even if aperture is the same).

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